2013-14 Colloquia talks

Abstract: In this talk, I will focus primarily on giving the motivation for my work in G₂- and multisymplectic geometry. I will begin with a brief introduction to general vector cross products and their relation to various geometries of current interest. I will then focus on specifically G₂-geometry, a 7-dimensional geometry based on 2-fold vector cross products, wherein I will also give a brief introduction to differential forms. Finally, we will put the pieces together, and I will discuss how all of this ties in to my research. Throughout, I will give glimpses of applications to various areas of physics.

Abstract: Action potentials are short-lasting events in which the electrical membrane potential of a cell rapidly rises and falls. Ca²⁺ signaling through ion channels is well understood at the plasma membrane, i.e, movement of Ca²⁺ ions from the extracellular space through Ca²⁺ channels to the intracellular space (cytosol). Ca²⁺ channels are also present on membranes of intracellular organelles such as the mitochondria and the ER. The channels control the flux of Ca²⁺ from intracellular stores to the cytosol. Cytosolic Ca²⁺ levels in turn regulate hundreds of cellular functions. While we know a great deal about Ca²⁺ regulated signaling pathways, we know much less about how other cations regulate cellular function(s). Recently Ning Xu et al. provided data indicating that intracellular Na⁺ may be modulated in a similar fashion to Ca²⁺. However, little is know about the regulation of intracellular Na⁺. Using well described ODEs, such as the Hodgkin Huxley and Nernst equations, we have explored how to model action potentials and how the release from intracellular stores may alter Na⁺ levels in the cytosol. Simulation results indicate that there is much we do not understand about the role(s) intracellular stores play in regulating cytosolic Na⁺.

Abstract: The colored Jones polynomial is one of the more fascinating objects in knot theory. We will show how to rediscover some well-known and less known q-Series Identities such as the (second) Rogers-Ramanujan identity from studying the colored Jones polynomial.

Abstract: Consider a population of individuals occupying some habitat, which is structured by age. Suppose that there are two distinct life stages, the immature stage and the mature stage. Suppose that the mature and immature population are not competing in the sense that they are consuming different resources, e.g. think of frogs and tadpoles. A natural question is "What determines the age of maturity?". In many biological contexts, the age of maturity is not merely constant but is more accurately determined by whether or not the quantity of food consumed by the immature population, reaches a prescribed threshold. In this talk we discuss some past work related to this idea, and new research directions it has inspired.

Abstract: Kostant's seminal paper "On Convexity, the Weyl Group, and the Iwasawa Decomposition" provides an elegant generalization of many results in matrix theory to the setting of a real semisimple Lie group. For example, there is a sense in which matrix spectra correspond to convex hulls of Weyl group orbits of elements of the semisimple group G, leading to a generalization of the Horn-Thompson theorem to the Lie group context.

Our research follows the spirit of Kostant's work. In this talk, we examine generalizations of well-known matrix results to the Lie group setting. In particular, we discuss two converging matrix sequences, the iterated Aluthge sequence and the iterated LR sequence, and examine the context in which these results may be generalized to the elements of a Lie group. We also discuss ongoing research on the inverse-adjoint decomposition and equivalent conditions. As in Kostant's work, we see that the available group decompositions play an important role in controlling group behavior, once again confirming that many well-known matrix behaviors are not so much due to the concrete matrix structure, but rather to the much deeper Lie group structure.

Abstract: Let F be a non-archimdean local field of characteristic zero. The local converse problem, an important problem in the Langlands program, asks to what extent the "twisted gamma factors" determine a representation of GL(n,F). Jacquet has formulated a conjecture on precisely which family of twisted gamma factors can uniquely determine a representation of GL(n,F). I will first give an overview of the Langlands program, and then report on recent progress on Jacquet's conjecture, which is joint work with Baiying Liu, Shaun Stevens, and Peng Xu.

Abstract: I will discuss several problems at the intersection of geometry and partial differential equations (PDEs), all of which arise from optimization problems. One can associate many measurements to a bounded domain D in n-dimensional Euclidean space, such as volume, perimeter, diameter, inradius, principle frequency, and torsional rigidity (which is also the maximum of the expected first exit time of a Brownian particle). An isoperimetric inequality, in the sense of Polya and Szego, seeks to extremize one measurement while holding another fixed. For instance, the classical isoperimetric inequality seeks to minimize perimeter while fixing volume, the Faber-Krahn inequality seeks to minimize the principle frequency while fixing volume (of all drumheads with an equal area, the round drum has the lowest base note). I will discuss several similar domain optimization problems associated to a nonlinear, second order PDE, which arises from extremal Sobolev functions and interpolates nicely between torsional rigidity and principle frequency.

Time allowing, I will also discuss a problem arising from conformal geometry on the sphere, which is associated to a fourth order PDE, as well as some interesting open questions.

Part of this is joint work with Tom Carroll, of the University of Cork in Ireland.

Abstract: We derive a new p-value based procedure which improves upon the Hommel procedure by gaining power as well as having a simpler step-up structure similar to the Hochberg procedure. The key to this improvement is that the Hommel procedure is nonconsonant and can be improved by a consonant procedure. Exact critical constants of this new procedure can be numerically determined and tabled. The 0th order approximations to the exact critical constants, albeit slightly conservative, are simple to use and need no tabling, and hence are recommended in practice. An analytical proof is given to show that the proposed procedure controls the familywise error rate under independence among the p-values. Simulations are performed to empirically exhibit familywise error rate control under both positive and negative dependence. Power superiority of the proposed procedure over competing procedures is also empirically exhibited via simulations. Illustrative examples are given.

Abstract: A scientific experiment typically addresses many research questions and therefore involves testing multiple hypotheses. The incidence of false positives can become very high if proper adjustment is not made for multiplicity of tests. Some researchers selectively report only positive findings from their experiments. This selective inference problem is also a multiplicity problem in another disguise. In this talk I will present examples of multiple testing from various published studies. Next I will discuss some simple statistical methods for multiplicity adjustment. Finally, I will conclude with some remarks about the need for reproducible and replicable experiments to confirm research findings.

Abstract: In this presentation, we shall examine some interesting properties of pedal triangles. A few of these properties are well-known in classical geometry, but many are observed and proved only in recent years. In particular, we pay attention to the sequences of pedal triangles, the fractals generated by pedal triangles, some extreme properties associated with pedal triangles and its relationship with the parental triangle. Basic multi-variable calculus can be applied nicely to the study of dynamical systems in classical geometry. Anyone with some knowledge in high school geometry and calculus should be able to appreciate the special role played by pedal triangles.

Abstract: The cross product of vectors in Euclidean 3-space can be introduced in undergraduate linear algebra courses. One might wonder why contrary to the dot product, which is available for any finite-dimensional real vector space, the cross product is only defined for 3-space. In my talk I will discuss the existence of cross products of vectors in Euclidean n-space for arbitrary n. It turns out that this is only the case if n = 0, 1, 3, 7.

This result is equivalent to a theorem of Hurwitz on the existence of real composition algebras (1898) and to a theorem of Zorn on the structure of finite-dimensional alternative division algebras over the real numbers (1932). Both kinds of algebras only exist in dimensions 1, 2, 4, or 8 (and are realized by the real numbers, the complex numbers, Hamilton's quaternions, and Cayley's octonions, respectively). All proofs are very elementary and can be done in a purely algebraic manner.

If time permits, I will also discuss a theorem of Hopf on the existence of finite-dimensional commutative division algebras over the real numbers (1940), for which no (elementary) purely algebraic proof is known. The goal of my talk is to present a lot of interesting mathematics from different areas (algebra, number theory, and topology) and the intimate relation between them.

Abstract: In this paper, we apply smoothed jackknife empirical likelihood (JEL) method to construct confidence intervals for the receiver operating characteristic (ROC) curve with missing data. After using hot deck imputation, we generate pseudo-jackknife sample to develop jackknife empirical likelihood. Comparing to traditional empirical likelihood method, the smoothed JEL has a great advantage in saving computational cost. Under mild conditions, the smoothed jackknife empirical likelihood ratio converges to a scaled chi-square distribution. Furthermore, simulation studies in terms of coverage probability and average length of confidence intervals demonstrate this proposed method has the good performance in small sample sizes. A real data set is used to illustrate our proposed JEL method. This is joint work with Dr. Hanfang Yang.

Abstract: In the final stages of a long and costly drug discovery process, a drug compound is introduced into humans through different phases of clinical trials. In oncology, as many as 60% of drug compounds that reach the last phase (Phase III) fail to show an improvement in survival. One possible reason for this high failure rate could be inappropriate evaluation of compounds in preceding Phase II trials, in which the primary endpoint is often binary tumor response measured by the Response Evaluation Criteria for Solid Tumors (RECIST). Extensive efforts have been ongoing to identify alternative tumor measurement (TM)-based endpoints, with the goal of improved prediction of overall survival (OS). We evaluate alternative categorical and continuous TM-based endpoints for their ability to predict OS using the RECIST 1.1 data warehouse (13 breast, lung, and colon cancer trials). We fit Cox models, using a landmark analysis to allow for early endpoint evaluation, and measure predictive ability via the c-index, Hosmer-Lemeshow goodness of fit statistic, positive/negative predictive value and prediction error. Absolute and relative change in TMs demonstrate potential, but not convincing, improvements in predicting OS compared to RECIST tumor response.

The talk should be accessible to a general audience. I will begin with a brief background to the problem, then describe our methods and results, and conclude with discussion and and open statistical questions.

This is joint work with Sumithra Mandrekar, Dan Sargent, Jeff Meyers, Axel Grothey, Xinxin Dong, Yu Han and Jan Bogaerts.

Abstract: The Basel problem, to find 1 + 1/4 + 1/9 + 1/16 + ... was open for a long period of time during the 17-18th century. Euler was the first one to give a beautiful although incomplete solution. We will discuss the problem and related questions.

Abstract: It is well known that there are homeomorphic 4-manifolds that are not smoothly equivalent, which become smoothly equivalent after taking the connected sum with one copy of a special manifold. Similar behavior may be found in other geometric settings, including diffeomorphisms up to isotopy, positive scalar curvature and knotted surfaces. In this talk we will prove that there is an infinite family of spheres in a connected sum of complex projective spaces with assorted orientations, so that no two spheres in the family are smoothly equivalent, yet every pair is topologically isotopic. Furthermore we will show that the spheres become smoothly isotopic after one stabilization. We will do so with an explicit description of the spheres and the isotopies. This is joint work with Danny Ruberman, Paul Melvin, and Hee Jung Kim.

Abstract: Ca2+ is an important regulator of various physiological functions, including modulation of vascular tone via the endothelium, a thin cell layer which lines the interior of blood vessels. Endothelium-mediated blood vessel relaxation is essential for maintenance of blood flow, where stimulation of Ca2+-dependent effectors produces robust dilatory responses. In endothelial cells, the Ca2+ permeable inositol 1,4,5-trisphosphate receptor (IP3R) is a major regulator of Ca2+ activity. Recent evidence suggests that IP3R-dependent Ca2+ activity leads to continuous attenuation of vascular tone in pig coronary arteries. This effect is amplified by stimulated production of inositol 1,4,5-trisphosphate (IP3), a phospholipid metabolite which increases IP3R activation. Fluorescent Ca2+ measurements using microscopy have revealed spatially and temporally complex IP3R-dependent Ca2+ signals, but critical aspects of their regulation by IP3 remain unresolved because real-time measurements of IP3 dynamics are not currently possible. However, mathematical modeling may be employed as a valuable tool in determining the impact of IP3R regulation on endothelial Ca2+ activity. Using a combination of fluorescence microscopy, novel calcium signal analysis techniques, and mathematical modeling, we tested the hypothesis that IP3 gradients and IP3R distribution determines the magnitude, speed, and duration of Ca2+ wave propagation in pig coronary endothelial cells.

Abstract: A totally positive matrix is a matrix all of whose minors are strictly positive. This class of matrices is of great importance in mathematics, from a theoretical and a practical point of view. In this talk I will explain some of the basic properties of these matrices, especially those concerning their eigenvalues. In particular, I will discuss the Gantmacher-Krein theorem, which states that the eigenvalues of a totally positive matrix are all positive and simple. Along the way, I will also discuss the Perron-Frobenius theorem on strictly positive matrices.

Abstract: Following works of G. Benkart, D. Britten, S. Fernando, V. Futorny, A. Joseph, F. Lemire, and others, in 2000 O. Mathieu achieved a major breakthrough in representation theory by classifying the simple weight representations of finite dimensional reductive Lie algebras. The next step in the study of weight representations is to look at the indecomposable representations. In this talk we will discuss some recent results related to the structure of the indecomposable weight representations and connections with algebraic geometry. This is a joint work with Vera Serganova.

Abstract: Strict polynomial functors were defined by Friedlander and Suslin in 1997 in order to investigate cohomology for the general linear group, and ultimately to prove that the cohomology ring of a finite group scheme (equivalently, a finite-dimensional cocommutative Hopf algebra) is a finitely generated algebra. In this talk I'll give an overview of what a (strict) polynomial functor is, and how these objects are connected to the cohomology of the general linear group. Toward the end of the talk, I will describe some questions I am currently investigating in a "super" analogue of the above setup. The speaker will endeavor to make this talk accessible to open-minded non-algebraists (and graduate students).

Abstract: It is not uncommon in mathematical research that different people prove the same theorem independently around the same period of time. It is also not unheard of for a mathematical result to be discovered, forgotten, and then rediscovered in the history of mathematics. Therefore, the attribution of a mathematical theorem to the appropriate author(s) often caused dispute or controversy, sometimes even involving unpleasant personal attacks. In this presentation, we shall take a closer look at the Napoleon theorem in classical geometry and its generalizations in modern mathematics. In addition to some interesting geometric, algebraic, and analytic results related to the Napoleon theorem and its generalizations, we will also review some mathematicians' personal experience as they challenged themselves with ambitious problems.

Abstract: For a finite graph Λ its spectrum is defined to be the set of eigenvalues of the adjacency matrix A_Λ counted with their multiplicities. In particular, it is possible to compute the spectrum for any explicitly given finite graph, but the isomorphism type of a graph is not determined by its spectrum.

In the talk it will be shown that for the Cayley graph Γ(G,S) of a finite group G with finite generating set S the representation theory of the group G yields a decomposition of the spectrum of Γ(G,S) in pieces which are related to the irreducible complex characters of G. This phenomenon has interesting applications in group theory and combinatorics.

Abstract: We say a square matrix has finite order if there exists a positive integer m such that the matrix raised to the m^th power is the identity matrix. A few weeks ago some of our most talented graduate students were experiencing sleepless nights trying to find integer matrices of finite order. Fortunately, someone in this picture will help us to avoid such nightmares in the future.

Abstract: We will begin be defining when two paths α and β are homotopic. Next, we will define the multiplication of paths, and the multiplication of equivalence classes. Namely, the equivalence classes of loops based at a fixed point, x₀, of an arbitrary surface X. The description of the fundamental group of the circle, S¹, will be given. Also, it will be shown that the fundamental group is indeed a group. If there is time, we will define covering spaces, by illustration, using the circle and torus. Lastly, we will ambitiously try to explore the relationship between covering spaces and the fundamental group. This talk is aimed at undergraduates and graduates who have not had a course in Algebraic Topology.

Abstract: PETER the Repeater
Platted round a platter
Slips of slivered paper,
Basting them with batter.

So begins "(Cats) Cradle Song," verse composed by James Clerk Maxwell, probably in 1877. It was Maxwell's response to the manuscript of "On Knots," written by his friend Peter Guthrie Tait, who had hoped for constructive criticism.

Maxwell wrote poems to amuse his friends and express personal sentiments. The purpose of "(Cats) Cradle Song" was to have a bit of good-natured fun at Tait's expense. Between its references to nursery rhymes, we discover mathematical ideas that were novel at the time. Some of the ideas would endure and inspire every succeeding generation.

Abstract: The idea of atoms dates back to ancient India (6th century BC) and Greece (5th century BC). Democritus coined the term atomos, meaning indivisible. We now know that atoms are divisible: physicists find smaller and smaller particles. In Mathematics points are indivisible. We will challenge this and define a new world where points are divisible and functions like signum(x) are continuous ... Please attend the talk only if you can sing "What a Wonderful World".

Abstract: In this talk, I will describe several embeddings or immersions of the projective plane in 4-dimensional space. The descriptions will follow from movie parameterizations and their associated charts. A movie is a sequence of 3-dimensional cross-sections of the surface while a chart is a transverse projection onto the plane of the blackboard. By including depth considerations in the chart, we can give full descriptions of the surface. In this way, a detailed understanding can be achieved.

Abstract: Lehmer's interest in large primes inspired him to construct a mechanical computer out of bicycle chains in 1928, when he was 23. It also led him to pose a deceptively simple question about polynomials with integer coefficients. Eighty years later, Lehmer's problem remains unsolved. It has turned out to have repercussions in graph theory, group theory, dynamics and topology.

Abstract: Virtual knot theory is introduced by L. H. Kauffman as a generalization of classical knot theory in the sense that if two classical link diagrams are equivalent as virtual links, then they are equivalent as classical links. I will talk about the history of the virtual links and introduce some invariants for classifying them.